I always just replace each embedded single quote with the sequence: '\'' (that is: quote backslash quote quote) which closes the string, appends an escaped single quote and reopens the string.
I often whip up a "quotify" function in my Perl scripts to do this for me. The steps would be:
s/'/'\\''/g # Handle each embedded quote
$_ = qq['$_']; # Surround result with single quotes.
This pretty much takes care of all cases.
Life gets more fun when you introduce eval into your shell-scripts. You essentially have to re-quotify everything again!
For example, create a Perl script called quotify containing the above statements:
#!/usr/bin/perl -pl
s/'/'\\''/g;
$_ = qq['$_'];
then use it to generate a correctly-quoted string:
$ quotify
urxvt -fg '#111111' -bg '#111111'
result:
'urxvt -fg '\''#111111'\'' -bg '\''#111111'\'''
which can then be copy/pasted into the alias command:
alias rxvt='urxvt -fg '\''#111111'\'' -bg '\''#111111'\'''
(If you need to insert the command into an eval, run the quotify again:
$ quotify
alias rxvt='urxvt -fg '\''#111111'\'' -bg '\''#111111'\'''
result:
'alias rxvt='\''urxvt -fg '\''\'\'''\''#111111'\''\'\'''\'' -bg '\''\'\'''\''#111111'\''\'\'''\'''\'''
which can be copy/pasted into an eval:
eval 'alias rxvt='\''urxvt -fg '\''\'\'''\''#111111'\''\'\'''\'' -bg '\''\'\'''\''#111111'\''\'\'''\'''\'''