This will check if you have a valid expression of the form ='something' or ="something else":

m/=(?:"[^"']+")|(?:'[^"']+')/;
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As you see we check for both valid possibilities (hence the use of |). This means that we do not have to save what was the first quote.

Here is a full explanation (with thanks to YAPE::Regex::Explain):

The regular expression:

(?-imsx:=(?:"[^"']+")|(?:'[^"']+'))

matches as follows:
  
NODE                     EXPLANATION
----------------------------------------------------------------------
(?-imsx:                 group, but do not capture (case-sensitive)
                         (with ^ and $ matching normally) (with . not
                         matching \n) (matching whitespace and #
                         normally):
----------------------------------------------------------------------
  =                        '='
----------------------------------------------------------------------
  (?:                      group, but do not capture:
----------------------------------------------------------------------
    "                        '"'
----------------------------------------------------------------------
    [^"']+                   any character except: '"', ''' (1 or
                             more times (matching the most amount
                             possible))
----------------------------------------------------------------------
    "                        '"'
----------------------------------------------------------------------
  )                        end of grouping
----------------------------------------------------------------------
 |                        OR
----------------------------------------------------------------------
  (?:                      group, but do not capture:
----------------------------------------------------------------------
    '                        '\''
----------------------------------------------------------------------
    [^"']+                   any character except: '"', ''' (1 or
                             more times (matching the most amount
                             possible))
----------------------------------------------------------------------
    '                        '\''
----------------------------------------------------------------------
  )                        end of grouping
----------------------------------------------------------------------
)                        end of grouping
----------------------------------------------------------------------
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Please note this does not work with forms such as var1='"1"' or similar.