I know one publisher's house style that asks for two variants:
"I'm gonna, gonna--" She sneezed."--be there in a minute, okay?" (Action stopped the dialogue here, so em dashes go inside dialogue.)
"I'm gonna"--she rubbed her nose--"be there in a minute." (She's talking and rubbing her nose here at the same time.)
Other publishers don't, and run with the the CMoS 6.84 guide or whichever house and style manual they're using.
6.84 explicitly says to put the dashes outside the quotes "if the break belongs to the surrounding sentence
rather than to the quoted material" (emphasis mine). Their only example, unfortunately, shows the case where the dashes are outside the quotes, but the wording of their rule strongly implies that the opposite should be done if the break
does belong to the quoted material. (That is, a reasonable reading of CMOS endorses the two variants you cite.)
Of the thousands (and, um, thousands) of books I've read, I've never in my life seen em dashes in a continuous piece of dialogue broken by an action beat placed inside the quote marks. Never. I understand there are different house styles, but I have apparently never come across a publishing house that does it that way. (Yes, sadly I'm a reader who pays attention to such things).
You might not be as attentive as you think. Besides Fallen's personal experience to the contrary (quoted above), I have definitely seen this in books from major publishers, such as, of what I've read most most recently, a Penguin Book published in 1978.
Anyway, for reasons already beaten to death upthread, in the particular example that started this thread, inside the quotes is the more logical place to put the dashes. Several have argued that either solution is correct, but I tend to feel that if one solution is correct but another is more correct, go with the latter.
In your example. the em dashes aren't quite necessary. a comma would have sufficed since it's less an interruption and more a pause.
The interrupting phrase is an independent clause, so a comma would create a comma splice.