Could silicon oil block Na+ ion channels in the membrane of an axon and prevent Na+ influx?
No, silicon oil couldn't block an ion channel. Oil is a nonpolar substance, just like the membrane lipids, so if you can get your oil to that cell, it will mix with the lipids within the membrane. The ion channel, which sits in the membrane, too, is rather closed to the membrane interior and opens to the inside and outside of the cell which, contrarily, is a polar solution of soluble molecules in water. So, to block one of the openings you better have a polar molecule that is water-soluble.

http://en.wikipedia.org/wiki/Ion_channel

http://en.wikipedia.org/wiki/Polarity_%28chemistry%29