In French, a canard is a duck.
In English, a canard is "an unfounded rumour or story".

What's the connection, and how did the word come into English?

On a quick check I saw that Compact OED explains, "French, ‘duck’, also ‘hoax’, from Old French caner ‘to quack’." Why would a hoax be equated with the quack noise?

Perhaps for the same reason that an incompetent or shyster is referred to as a "quack"?

quote:

Perhaps for the same reason that an incompetent or shyster is referred to as a "quack"?

In the UK that expression is generally reserved for those in the medical profession - usually doctors.

I'd say that we use quack solely doctors. For example, you wouldn't refer to a quack nurse.

I'd think shyster is only for a lawyer.

Yes, in the medical field, from my experience, it is only the physicians who are called "quacks." I often wondered where that word came from.

Well I figured any snake oil salesman equivalent could be a quack, but the dictionary agrees that it should be a medical charlatan

From the online etymology dictionary...

quack (v.)
"to make a duck sound," 1617, quelke, of echoic origin (cf. M.Du. quacken, O.C.S. kvakati, L. coaxare "to croak," Gk. koax "the croaking of frogs," Hitt. akuwakuwash "frog"). M.E. on the quakke (14c.) meant "hoarse, croaking."
quack (n.)
"medical charlatan," 1638, short for quacksalver (1579), from Du. kwaksalver, lit. "hawker of salve," from M.Du. quacken "to brag, boast," lit. "to croak" (see quack (v.)) + zalf "salve." Cf. Ger. Quacksalber, Dan. kvaksalver, Swed. kvacksalvare.