There is much information at
Axolotls: The Fascinating Mexican Axolotl and the Tiger Salamander. If you haven't already read that, it's a good idea - specifically the Housing and Requirements pages.
10 gallon is considered the minimum size for one axolotl, so you're good there. As for the filter, worry about flow; oxygenation most likely won't be a problem...
...And on that note, he's probably not short on oxygen, but rather likely has concerns with the water quality. Are you doing partial water changes on a regular basis? You probably should be changing out ~20% of the water daily with new
dechlorinated water until the tank cycles (axolotl.org talks about tank cycling, and a search here will reveal more) - or at least that was my experience.
You can test the water parameters with a water quality with a test kit like
this one from API, or with separate test kits for each parameter. You should also be able to have your local fish store test the water for you - though it seems they prefer to just provide "it's fine" as an answer, rather than the actual results of the test, so you may have to ask specifically for the numeric results.