Bitten by regular expression syntax.

next if $FILE =~ /^./;
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Should be:

next if $FILE =~ /^\./;
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In your match you were looking for each filename starting with, not a 'dot', but any single character. Naturally everything matched. Easily done.

Update:
Turns out you have a few other issues. As Corian says, you need to prefix $FILE with the directory name. There is another issue as well - for some reason you require a reference to an array to be passed to your subroutine (why?). Here are the relevant lines modified:

if(-d "$FILEPATH/$FILE"){             
               #Don't wanna have current directory and the one below
               next if $FILE =~ /^\./;               
              
               push(@FILELIST,@{readin_directory(["$FILEPATH/$FILE"])}
+);            
            }
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I have not tested this on a wide range of naming conventions. Notice that the argument to the subroutine call (& not required) is in square brackets, to generate a ref to an array.