Stanley Wells's edition of the play (The New Penguin Shakespeare, 1972) has the following gloss for "bogs":
bawdy; it is not certain whether bog meant "privy" in Shakespeare's time
R. A. Foakes's edition of The Comedy of Errors (The Arden Shakespeare, Routledge, 1962) does not gloss "bogs" at all.
A Shakespeare Glossary by C. T. Onions (revised by Robert D. Eagleson, Oxford University Press, 1986) has no entry for "bog".
Eugene F. Shewmaker's Shakespeare's Language: A Glossary of Unfamiliar Words in His Plays and Poems (Facts on File, 1996) has the following entry for "bogs":
ref. to the bogs in Ireland; here in poss. wordplay with slang sense of "latrine": "Marry, sir, in her buttocks; I found it out by the bogs." Errors, III, ii, 115-116.
However, W. W. Skeat's A Glossary of Tudor and Stuart Words, Especially from the Dramatists (completed by A. L. Mayhew and published at the Clarendon Press in 1914) has the following entry for "bog":
a privy, latrina, Shirley, Witty Fair One, iv. 6 (end).
And, indeed, James Shirley's play The Witty Fair One contains the following lines (quoted from The Best Plays, 1888):
[A cry within] A Teague! a Teague! Make way, for shame!
Mis. Bonavent: I think they are started.
The two Runners cross the stage, followed by Lord Bonavent, Venture and others.
[Exeunt all but Mis. Bonavent and Mis. Carol.]
Mis. Bonavent: He may be in the bog anon.
Mis. Carol: Can they tell what they do in this noise?
Pray Heaven it do not break into the tombs
At Westminster, and wake the dead.
The Witty Fair One dates from 1628, just over a decade after Shakespeare's death and it is not entirely obvious that "bogs" here means "latrine".
Following a hint from Gareth Rees, I found "boggard" in the Huloet's Abcedarium anglico latinum, pro tyrunculis Richardo Hulœto exscriptore, printed in 1552, but I found that entry too difficult to decipher (except for the last word "Siege").
However, Huloets dictionarie newelye corrected, amended, set in order and enlarged ... by Richard Huloet, printed in London in 1572 has the following entry for siege, which lists "bogard" as a synonym, making it plausible that Shakespeare used "bogs" also in this sense:
Siege, iacques, bogard, or draught. Latrina, ae, Fo∣rica, ae. f. g. Vn priue, ou retraict. S.