I am assigned to an Epic expected to take 3 developers for 6 weeks.
There are so very many things wrong with that statement, from a Scrum perspective. I'll go through them in order.
I am assigned to
Scrum works better as a pull-model, not as a push-model. No one assigns work to developers. In the Planning Meeting, the Scrum Team accepts work from the backlog (with the Product Owner prioritizing which work is most important, and the Development Team deciding how much they can get done that Sprint). During the Sprint, developers will, on their own initiative, pull down unassigned work and start working on it.
Epic expected to take
Epics are not estimated. They are far too large and nebulous to estimate accurately, so attempting to do so does nothing but give a false sense of schedule-security.
3 developers for 6 weeks.
Two things wrong here. First, estimates should be done in terms of relative effort (story points), not time. Second, instead of estimating in x-points-for-y-developers, just estimate for x-points. Otherwise, you are locking yourself into how many developers will work on the Epic, which is distinctly un-Agile.
Now, to answer your actual question,
When is the best time for developers to break Stories down into Tasks?
The answer depends on whether or not you are actually doing Scrum. If you are, then the answer is during the Sprint Planning Meeting (or, as some Teams do, during the Pre-Planning Meeting).
However, it's looking like your company does not really do Scrum. The commonly accepted word for what you're doing is Scrumbut. So your two options are:
Try to convince your company to switch to actual Scrum. If the reasoning behind this particular version of Scrumbut was something along the lines of:
"This Scrum thing looks nice, but we probably don't need this, and this, and this..." then take a look at We Tried Baseball and It Didn't Work.
Ask your company how they intend to implement their particular version of Pretends-To-Be-Scrum-But-Actually-Is-Not-Even-Agile. Because we, not being part of your company, cannot really answer that question for you. Keep in mind that it's possible there might not be anything actually wrong with what your company is doing, from a business-perspective... but calling it Scrum is blatantly false. As the Scrum Guide notes:
although implementing only parts of Scrum is possible, the result is not Scrum.
One final thing, you might want to look into how you're defining User Stories. A commonly-used format is "As (identity), I would like to (action) in order to (reason)." See the INVEST mnemonic for more details.