Does NY State (upstate, not NYC) have any restrictions on rifles & shotguns?
In a word, "yes". The NYS AWB (section 22 here: http://law.onecle.com/new-york/penal/PEN0265.00_265.00.html) has a provision that limits post-ban semiautomatic shotguns which may restrict a magazine tube extension, depending upon the configuration of the shotgun. Pump shotguns are excluded.
Note that other upstate cities have their own AWBs as well. I know Rochester does; I'm not sure about Buffalo, Albany & Yonkers, but the relevant page on the NYSRPA website (http://www.nysrpa.org/nygunlaws.htm) suggests that there is something worth researching.