I'm going to start a campaign using some of the variant rules in Uneartherd Arcana. One of my players wants to use the necromancer variant (Unearthed Arcana pg. 63), and I'm not sure if the skeleton minion gets feats normally or not.
This is a replacement for the familiar class ability of a wizard, and many of the qualities of the minion are similar to familiar abilities. Though the text does not explicitly say so, I would be inclined to treat the minion's power progression similar to a familiar. That would exclude gaining a feat per 3 HD. However, given that all the other familiar abilities (besides HD increase, natural armor bonus, and Str/Dex increase) are not present with the minion, I would also consider allowing the normal feat progression based on HD. There seems to be no rules text for or against it. So, I would say it is up to you.
Edit: In looking at the HD-progressed skeleton examples in the Monster Manual, the skeleton template specifically removes all feats from the base creature and grants only Improved Initiative. So, it seems like "Improved Initiative and no other feats" is a special quality for skeletons in general (perhaps due to @KRyan's reason in his answer). A skeletal minion would seem to follow suit. So, I would say no feats based on HD, but add Improved Initiative as a bonus feat acquired via the skeleton template.