You can only choose a soldier's specializations if you activate the respective Second Wave Options, either prior to the campaign or through the difficulty menu. There's also an SWO that allows you to choose the Squaddie (or origin) perks.

The default behavior is that each soldier gets 2 random perks on Squaddie as well as access to all specializations.

Oh that makes sense now! I think I figured it out, looks squadie level skills are actually tied to the weapon you select?

That's not quite correct. Some weapons themselves grant abilities, e.g. Squadsight on the Sniper Rifle, Aid Protocol on the Gremlin etc. The Squaddie abilities are completely independent.