Subordinating conjunction following conjunctive adverb

As I thought, the correct sentence was:

Dogs are usually friendly; however, while eating, some are unpredictable.

See http://www.chompchomp.com/terms/conjunctiveadverb.htm for more on conjunctive adverbs.

The conjunctive adverb ‘however’ is really completely irrelevant here.

The relevant issue is that “while X-ing” acts as a sentence adverb, and sentence adverbs are (almost) invariably followed by a comma. If we recast the sentence to move the conjunctive adverb away from the start of the sentence, a comma is still preferable:

Dogs are usually friendly. While eating, some are unpredictable, however.

In certain cases, the comma may even disambiguate an otherwise ambiguous sentence (as pointed out by Annick—note that I have no idea if the following are anatomically correct or not):

While eating, the brain speeds up your metabolism.
While eating the brain speeds up your metabolism(, eating the tongue does not).

The latter means something entirely different from the former!