Squeezing and drying hair on Shabbos:
Biblically, there is no prohibition against squeezing hair on Shabbos. Nevertheless, it is Rabbinically forbidden to squeeze liquid out from hair on Shabbos [due to the Mifarek prohibition]. Thus, every bather needs to beware not to squeeze his hair.
Q&A on squeezing and drying hair on Shabbos
What areas of hair contain the squeezing prohibition?
Only those areas which have hair growing in close proximity to each other such as the hair of the head, beard [underarms and pubic area]. However, the hair of the arms [and legs] are distanced from each other and thus do not contain a squeezing prohibition.
May one squeeze the water out from his eyelids after leaving the Mikveh?
This matter is dependent on the dispute mentioned above. Practically, one may be lenient to do so as otherwise there is pain involved in having the water get into one’s eyes, as well as that since the hair of the eyelids are not close together they most probably do not contain a squeezing prohibition at all.
If while drinking one wet his mustache, may he squeeze the water out?
One is to be careful to only do so using a towel as opposed to one’s hands.
May one scratch his hair when it is wet?
Yes. Although doing so will inevitably cause water to get squeezed out it is nevertheless permitted to be done. Nevertheless, it is best to do so using one’s hat or the like.
 Admur 326:6; Raba and Rav Yosef in Shabbos 128b
The reason: As hair is hard and does not actually absorb water inside of it. [Admur ibid; Rashi ibid] The water is absorbed between each individual hair. It hence does not carry a squeezing prohibition of Mifareik of laundering which is relevant to clothing and other items that are squeezed.
 Admur 326:6 and in 330:1; M”A 326:8 and 320:23; Beis Yosef 330; Kesef Mishneh 2:11; Implication of Rambam 9:11; Kol Bo and Maharil 139 brought in M”A ibid and Beis Yosef Y.D. 199; M”B ibid and 326:24, 25, 320:55; Biur Halacha 302 “Assur”; Aruch Hashulchan 320:35; Kaf Hachaim 320:110
Other opinions: Some Poskim rule there is no prohibition involved in squeezing hair. [Implication of Gemara ibid and Meiri Nidah 67b; Rashi and Rabbeinu Tam according to Avnei Nezer 157:14; Chida in Shiyurei Bracha 199:3 based on Beis Yosef Y.D. ibid; See Imros Moshe p. 332 for a lengthy discussion on the opinions on this matter.]
Does the laundering prohibition apply to hair? No. [Avnei Nezer ibid]
 Admur 326:6; M”A 326:8; M”B 326:21
 Ketzos Hashulchan 133 footnote 8; Piskeiy Teshuvah 326:7; Shabbos Kehalacha 18:53
 Ketzos Hashulchan ibid; Minchas Shabbos 86:6; Rav SZ”A in Maor Hashabbos 4:44; Shabbos Kehalacha ibid
 Yeshuos Chochmah brought in Ketzos Hashulchan ibid; Az Nidbaru 13:10
 Yeshuos Chochmah brought in Minchas Shabbos ibid; Az Nidbaru 13:10; Rav SZ”A in Maor Hashabbos 4:44;
 Piskeiy Teshuvos 326 footnote 133
 Ketzos Hashulchan 133 footnote 8
 And in a case of pain the Sages did not uphold their decree.
 Ben Ish Chaiy Pekudei 2:8; Ketzos Hashulchan 133 footnote 8
 Minchas Shabbos 86:6
 The reason: As a) One does not desire the water to get squeezed and it is thus a Pesik Reishei which is not Nicha Lei. As well b) In a case of pain the Sages did not apply their decree.