If I use Aegislash's "Stance Change" Ability, to switch for another Aegislash, can I then use the new Aegislash's "Stance Change" if I want to?
No, even though you have switched to a new Aegislash card you cannot use Stance Change again because the effect of using it remains on that Pokemon.
Source: XY FAQ (2014-02-06), TPCi Rules Team (2014-02-06)

Relevant cards

Note: All applicable cards may not be listed below.
XY1_85
XY1_86