How do I ignore ownership on an external hard drive without using Get Info?
The partition that Xcode is installed on must be formatted for OS X. This will require reinstalling Xcode after formatting. Be sure to save any important files from the drive elsewhere. 

In Disk Utility, choose the partition and select erase.

Format the partition then reinstall Xcode. 

This should resolve the ownership issue. 

It appears from your screenshot that the partition cannot be modified.  You mention in your comment that the partition "has all of the external drive memory allocated to it."  Not sure what you mean here, are you using it for other purposes too?  If not, and you don't have another drive to use, in order to use this drive instead of the default installation on your main drive, it seems you have to reformat the drive.