Does the validity period of an re-entry permit count towards the requirement for naturalization?

No, a re-entry permit merely permits you to apply for admission and the extended absence will still delay your eligibility to apply to naturalize.

Yes, only the time in the U.S. counts toward naturalization and breaks longer than 180 days usually break continuous residency.

An absence of 6 months or more is likely going to cause a break in continuous residence for naturalization purposes. The use of a reentry permit is generally irrelevant. You can understand the naturalization requirements without a lawyer, if you wish, by reading the policy manual very closely.

Your re-entry permit will not preserve your physical presence/continue residence for purposes of naturalization. An absence of more than 180 days could potentially break the continuous residence requirement, forcing you to start over and having to apply four years and one day after your latest return from an extended absence.

If you leave the U.S. for over six months, you should assume that you have broken the continuity of your residence and your five years starts all over for naturalization.