It would be very unlikely to sink a ship. All wooden ships took on water, and had pumps aboard to pump the water out again. In battle, ships would get hulled below the waterline by cannonballs, and the crew could still fother the ship (block the hole with tarred canvas), then repair the hole, all while pumping out the excess water to stay afloat. So a lad with an auger would be unlikely to cause a ship to founder quite that quickly!

Having said that, if he cut out a large enough hole, he could indeed sink the ship. A saw would be a better tool for this than an auger, and it would take him a long time, during which (if he did not freeze to death) he would be vulnerable to musket fire, so it would not be a useful method in battle.