But when the regexp object qr/foo/ is instead assigned to the variable $exp via the get method, it (seems) to cease to be a regexp object. In fact, it seems to become the string literal 'qr/foo/'. So what is the get method doing to my regexp object and how can I make it stop?

You're hitting the nail on the head, but not seeing it, I think. When you type "qr/foo/" into your entry widget, it is just a string. It never "ceases" to be a regular expression object - it never was one. You might have typed anything into your text box, a limmerick or a dirty word or anything. Just because the text is "qr/foo/", and happens to be a string that stands for a regular expression object in the Perl language is immaterial. The program won't look at the text and say, aha, this is a perl construct. It's just text.

This is completely different from writing my $exp = qr/foo/; in your perl script. This is part of your program (and not your program input), and perl knows how to parse this. If, instead of my $exp = qr/foo/; you wrote my $exp = "qr/foo/";, you would get the same behavior as when the regex is passed via your text entry widget.

If you want Perl to parse and interpret the entered text as a chunk of Perl code, you need to use the string version of eval. That tells perl that the text passed in is more than text, and to evaluate it as a Perl expression. I hope this makes sense.

As for allowing a user several ways to pass a regular expression, you should read the text entered and adjust your processing accordingly, like [id://injunjoel] suggests. Something like this (completely untested):

my $regexp = undef;
my $entered_text = $regexp_entry -> get();

if ($entered_text =~ m!^qr/!){
   # User has passed a "qr//" type regex:
   $regexp = eval $entered_text;
}
else{
   # User has passed a straight regular expression
   $regexp = m/$entered_text/;
}
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Note that there is no need in the "else" branch for the user to have enclosed their code in single- or double-quotes. You are reading text from the widget, then using that text to construct a regular expression at run time.