Can I enclose my new boiler in a closet?
Given that setup, I can't think of any reason you'd need to draw inside air. On the other hand, I'm not sure you want it entirely at outside temperature, from the point of view of freezing risk when not circulating. (Then again, I'm in a northerly climate; you may not be.)

Did the folks who installed it have any opinion on this approach?

Make sure there's enough space around the unit to (a) meet its specifications for fire safety and (b) allow servicing it and its support plumbing/pumps/whatever without having to rip down and rebuild that closet.