WRB: I really had intended to answer your question better. I, however, ran out of time. I was meeting a couple of Ham friends for lunch and had about a 1-hour drive to get there.
Let me now continue. The end-fed wire configured in a loop type set-up as you were asking about might work. The thought that comes to mind with me is, the impedance of that antenna "bent" in a loop is going to be altered so dramatically that it will increase your feedline loss I believe considerably.
Allow me to explain. The matching transformer in the EF wire is structured to present a low VSWR on multiple frequencies in the Amateur bands. This type of antenna is structured to be in a straight line, inverted V or inverted "L" configuration. When used in these configurations the "Balun" fools the transmitter and feedline into thinking the wire is resonant on multiple frequencies.
Nothing magic, just what it does.
Now to your question. I believe bending that antenna in a loop will alter the impedance so drastically that your coax feedline in any length greater than 50 ft long will see an extremely high mismatch that it will create signal loss even on receive.
However, this is just my opinion. I feel there are better solutions. One thought is to remove said balun and attach your coax feedline directly to your square/rectangle/triangle configuration, whichever suits your new location best. The shape of this "Loop" for receiving is certainly not critical in the slightest. However, the ultimate goal is to get the antenna to "Hear" as many frequencies well and with minimal loss at the lowest frequencies, this may take some more length. (IF possible!)
This leads me to a question. What are your primary interests in receiving "well"?
Next, is what is your receiver(s) / (equipment)?
Sorry not trying to cloud the issue, just trying to give you my thoughts on what may be the best solution for your new location, plus give you the best results for your listening interests.
I will tell you this, a horizontal "loop" type configuration is really a good choice as most generally a loop will receive "less" man-made noise than the EF antenna in any configuration.
The loop also as you suspected "tend" to be more Omni-directional.
The Horizontal EF antenna "tends" to be best in the directions following the "ends" of the antenna.
All the Best
Gary/W9FNB