Gemara

(Beraisa): We may squeeze plums, quinces and crabapples, but not pomegranates;

In the house of Menasiya bar Menachem they would squeeze pomegranates.

Since these other species are not normally squeezed, one may squeeze them l'Chatchilah.

(Rav Nachman): The Halachah follows this Tana, for he holds like Menasiya.

Objection: Menasiya is not the majority of the world!

Answer: Indeed, the practice in one region can determine the Halachah everywhere!

(R. Chanina): R. Elazar forbids growing cactus in a vineyard because in Araviya people grow it for their camels.

Rejection: Araviya is a whole region. Menasiya is an individual. His opinion is Batel to that of normal people.

Answer: Rav Nachman holds like Rav Chisda;

(Rav Chisda): If one squeezed spinach and put the juice in a Mikveh, if it changes the appearance of the water, the Mikveh is Pasul.

Question: People do not normally squeeze spinach!

Answer #1: Since he considers it important, it is considered a liquid.

The same applies to squeezing pomegranates!

Rejection (and Answer #2 - Rav Papa): Spinach juice disqualifies a Mikveh because one may not make a Mikveh from it.

145a (Rav): One may squeeze pickled vegetables for their sake, but not for the liquid. B'Di'eved, he is exempt.

One may squeeze cooked vegetables even for the liquid.

(Shmuel): One may squeeze pickled or cooked vegetables for their sake, but not for the liquid. B'Di'eved, he is exempt.

(R. Yochanan): Whether they are pickled or cooked, one may squeeze for their sake, but not for the liquid. B'Di'eved, he is Chayav Chatas.

Question (Beraisa): One may squeeze pickled vegetables needed for Shabbos, but not for Motza'ei Shabbos;

One may not squeeze olives and grapes. B'Di'eved, he is Chayav Chatas.

Answer (for R. Yochanan): It permits squeezing pickled vegetables needed for Shabbos, but not for Motza'ei Shabbos. The same applies to cooked vegetables;

This is for their sake, but not for the liquid. B'Di'eved, it is as if he squeezed olives and grapes. He is Chayav Chatas.

Rishonim

Rif (60b): All agree that a liquid that goes into a food is like a food, so one may squeeze onto food.

Rambam (Hilchos Shabbos 21:12): One may squeeze other Peros, such as quinces, apples and crabapples, for squeezing does not apply to them.

Rosh (22:2): All permit squeezing l'Chatchilah on Shabbos other Peros, e.g. plums, quinces, crabapples and similar fruits.

Poskim

Shulchan Aruch (OC 320:6): One may squeeze lemons.

Beis Yosef (DH Tutim): Rashi explains that squeezing Peros is Mefarek, a Toladah of Dash. Regarding Peros not normally squeezed, such as plums, Dash does not apply. R. Yerucham says that Rashi forbids squeezing any fruit. We concluded that Rav Nachman forbids squeezing pomegranates due to Rav Chisda's reason, that squeezing them shows that he considers them to be a liquid. They become a liquid for him. Rav Nachman explains that the Beraisa permits squeezing plums... to sweeten the fruit, but not for the liquid. One may not squeeze pomegranates even for the fruit, since Beis Menasiya used to squeeze it for the liquid. Beis Menasiya are not the majority of the world. However, if we permit squeezing to sweeten, perhaps people will squeeze for the liquid. Since he considers it a liquid, it is a liquid for him, and he is liable. We are not concerned for plums, since no one makes it a liquid. Tosfos (144b DH Hachi) says so. The Hava Amina was that one may squeeze them properly (for the juice). In the conclusion, we cannot say so, since he considers it a liquid. The other Poskim permit squeezing. Rav Chisda does not mean that since he considers it a liquid, it is a liquid for him. Rather, it is like Rav Papa said, since one may not make a Mikveh from it l'Chatchilah. Based on this, we can explain squeezing simply (i.e. for the juice). Therefore, we hold like this. We can say that there were others like Beis Menasiya. The Beraisa mentioned one such person. What many consider important is important, even if they are not the majority of the world, like Araviya.

Beis Yosef (ibid.): I equate everything squeezed for its juice to berries and pomegranates. The Rosh (Teshuvah 22:2) connotes like this. He said that lemons are like other Peros, which one may squeeze, for it is not common at all to squeeze them for liquid. They are squeezed only for food, so one may squeeze them on Shabbos. This connotes that if it were common to squeeze them for the liquid, it would be forbidden, like for berries and pomegranates. In the days of the Gemara they made apple wine (Avodah Zarah 40b). Surely it is more stringent than pomegranates, from which they did not make wine. We said "Menasiya's opinion is Batel to that of normal people." Why weren't apples mentioned? Surely they are like berries and pomegranates. We do not require that most people squeeze them for liquid. It suffices if we know about one place. However, they must drink the juice for thirst or pleasure, but not for Refu'ah. People squeeze plums, lemons and pomegranates for Refu'ah [and even so, the Gemara, Rosh, and the questioner, who asked why we learn from an individual, permitted them]. However, in Mitzrayim they squeeze lemons into water with sugar, and drink it for pleasure! They do so even on Shabbos, and no one protests. Perhaps it is forbidden only if the juice is drunk intact, without mixing it. Alternatively, one may not squeeze the juice alone, and later mix it, but one may squeeze it into another liquid. In any case one may squeeze into a food, even for grapes.

Taz (5): The primary Heter for the custom in Mitzrayim is like the Beis Yosef brought from Shibolei ha'Leket, that all know that it is to sweeten the food, and not for the liquid [so we permit even into an empty bowl]. Also here, he intends only to add taste to the water.

Magen Avraham (8): We permit even those who squeeze lemons into water and drink it, since it is not normal to drink the juice alone.

Mishnah Berurah (22): Nowadays, they fill hundreds of barrels with lemon juice to drink with punch. It is not clear whether one may squeeze them on Shabbos into liquid. Perhaps it is like berries and pomegranates, since they are squeezed into Kelim by themselves. Therefore, one must squeeze them onto sugar, and then it is like a liquid that goes into a food. Surely one may squeeze them onto food.

Bi'ur Halachah (DH Lischot): The Rosh permitted squeezing lemons for the juice. Surely he did not need to permit squeezing onto food. This is permitted even for grapes! Regarding lemons, one need not be concerned for the stringency of R. Chananel.

Kaf ha'Chayim (36): Mateh Yehudah permits squeezing lemons for liquid, and even to drink it afterwards. Neveh Shalom permits squeezing for liquid only if afterwards he will add it to food. The Radvaz says 'perhaps the Rosh permitted, for in his days they did not make lemonade, but in Mitzrayim they fill barrels of lemon juice and sell it elsewhere. This is like pomegranates of Beis Menasiya. Therefore, I squeeze onto sugar before adding water.' Mor u'Ktzi'ah rejects the Beis Yosef's Heter to squeeze them for liquid. Therefore, one should squeeze onto sugar.

Kaf ha'Chayim (37): One who wants to eat eggs with lemon juice should cut the egg, and squeeze onto the pieces.

Shemiras Shabbos k'Hilchasah (ha'Chadash, 5:2, footnote 11): One may not squeeze a lemon slice by hand or with a spoon, even when it is in a liquid. The Mishnah Berurah (320:34, Sha'ar ha'Tziyun 318:146) says that the Shulchan Aruch is stringent about squeezing snow into water because this is like squeezing Peros. All the more so, we forbid here!

Rema (1): Where the custom is to squeeze a certain kind of fruit to drink the juice for thirst or pleasure, it is like berries and pomegranates. If they squeeze only for Refu'ah, there is no concern.

Ashrei ha'Ish (2:28:1,3, citing ha'Gaon R. Y.S. Elyashiv Ztz"l): One should be stringent about citrus fruits destined to be squeezed for liquid, like for olives and grapes. Perhaps most are planted for this. If one cut a lemon in order to put it into tea, this is for the sake of liquid, so any juice that comes out [before putting it in the tea] is forbidden.