Hi bretth, and welcome to the forum 
I wasn’t saying it would remove dropbox on most Linux systems … but Peppermint has it’s own dropbox integration for the LXDE desktop which uses the pcmanfm file manager … directly from the Peppermint specific PPA … and in this case, YES Synaptic, Software Centre, and apt-get wants to remove it.
if you want to take a look at the peppermint-dropbox_3.0.0_all.deb, you can find it here:
http://ppa.launchpad.net/kendalltweaver/peppermint/ubuntu/pool/main/p/peppermint-dropbox/
It’s simply that there’s no version for the LXDE desktop with pcmanfm as the file manager … so I tried the “Generic” version.
(it’s quite possible I could change to using Thunar, and use the Xfce version, then install dropbox in the normal fashion … but I like pcmanfm)
IIRC, I used the “Generic Installer” version (which your site says - “use this if you’re not using Cinnamon, GNOME Shell, KDE 4, MATE, or Xfce”) which wanted to install nautilus which conflicts with peppermint-dropbox … but I could be wrong … if you want me to try again, and tell you exactly what the issue was, let me know 
[EDIT]
mark@AA1-Blue ~ $ sudo apt-get install insync-beta-ubuntu
Reading package lists... Done
Building dependency tree
Reading state information... Done
The following extra packages will be installed:
brasero brasero-cdrkit brasero-common dvd+rw-tools gir1.2-nautilus-3.0
gnome-desktop3-data growisofs hwdata libbrasero-media3-1 libexempi3
libgmime-2.6-0 libgnome-desktop-3-2 libquvi-scripts libquvi7
libtotem-plparser17 [b]nautilus[/b] python-nautilus wodim
Suggested packages:
vcdimager dvdauthor readom cdrskin cdrdao eog gnome-sushi cdrkit-doc
[b]The following packages will be REMOVED
peppermint-dropbox[/b]
The following NEW packages will be installed
brasero brasero-cdrkit brasero-common dvd+rw-tools gir1.2-nautilus-3.0
gnome-desktop3-data growisofs hwdata insync-beta-ubuntu libbrasero-media3-1
libexempi3 libgmime-2.6-0 libgnome-desktop-3-2 libquvi-scripts libquvi7
libtotem-plparser17 [b]nautilus[/b] python-nautilus wodim
0 upgraded, 19 newly installed, 1 to remove and 0 not upgraded.
Need to get 10.3 MB of archives.
After this operation, 30.7 MB of additional disk space will be used.
Do you want to continue [Y/n]?
It seems peppermint-dropbox contains an executable script which it installs at /usr/bin/nautilus that just contains:-
#!/bin/bash
pcmanfm $1 $2 $3 $4 $5 $6
So if Insync wants to install the full nautilus that would be overwritten, destroying calls to nautilus being passed to pcmanfm instead.
so they’ve added:-
Conflicts: mint-dropbox, nautilus
to the peppermint-dropbox DEBIAN control file.
I wasn’t saying it was an insync “issue” as such … just that it’s not for Peppermint/LXDE/pcmanfm 
.