What are the leading or trailing days on a monthly calendar called?

wraparound, especially used in the sense of:

an object that encircles or especially curves and laps over another

It fits pretty well into the sentence:

When viewing the month version of the calendar online, users can turn on/off the view of wraparound days from other months.

This is not a direct answer to the question as posed, but it does address Op's 'example sentence' if that is really what is required.

I suggest that:

• you do not need a descriptive term at all for such days: you can simply refer to them as "days from other months";

• "adjacent months" would be clearer that "other months";

• "dates" might be better than "days";

• "toggle" or "toggle the display of" might be better than "turn on/off the view of".

Additionally:

• I dislike "month version" and would suggest "month mode".
• Do you need to include "online"? How else will they be viewing it?

So you get:

When viewing the calendar in month mode, users can toggle the display of dates from adjacent months.

[This answer was initially written in response to an earlier version of the question. A number of the suggestions made in this answer have since been incorporated into the question.]