GFCI - should my neutral and ground have continuity?
At least violate Code a little less

Putting a GFCI in a box on a cable is a codevio. Using Romex for cordage is a codevio. And using junction boxes for portable boxes is also a code vio, but let's at least use a tough box and a strain relief, eh?  Here's what you need. 


square steel junction box, 4" square, drawn one-piece (not welded 5-piece) 
Strain relief that fits the proper cordage you go out and get right now (get 12/3 since in cordage, ground counts). I can't bring myself to link the cable clamp for Romex, because it's so wrong... 
4" square 1-device mud ring with at least 1/2" depth (plays better with Decora) again must be drawn not welded
Metal Decora cover plate - cut the screws short if needed  (Decora = large rectangular opening)


Alternate: a drawn steel Decora junction box lid can replace the last 2. Even though it's tougher, I avoided it because you'll have to bend/break off the Decora cover plate ears, and that'll wreck the GFCI for use anywhere else. 

Alternate: if you want to put a GFCI and feed another plain outlet for 4 sockets then use a 4-11/16" box and mud ring, otherwise it won't all fit. You can use a plain receptacle and normal/Decora split cover plate.














GFCI's, thru continuity, and ground

Safety ground is continuous through a GFCI.  

LINE neutral to LOAD neutral is not continuous. 

LINE hot to LOAD hot is not continuous. 



As you can see, both hot and neutral go through the mysteryworks of a GFCI device.  (actually, that includes a set of relay contacts, and also a current-sensing inductor, so you may read an impedance near zero.)  Needless to say, if the relay contacts are open, line-load will read as dead open...  IIRC the GFCI also has some electronics between LINE hot and LINE neutral, so you may also expect some non-infinity impedance there. 

Now, look close:  you see that green "upside down T" where the ground wire branches into the GFCI? No, you do not see that "T"?  There's a reason you don't: GFCIs don't connect to ground.  GFCI receptacles do, but only for the sake of the receptacle sockets; the GFCI portion doesn't use it.