I did not find your code particularly easy to understand.
An alternate coding could be like the following...
tr is a lot faster than using the regex engine.

#!/usr/bin/perl
use warnings;  
use strict;

my @words = ("anyone", "cancel", "declArE", "perlmonks", "pal");
# Note the "pal" case is an extra case that I added

foreach my $word (@words) 
{
   if ($word =~ tr/aA//) # word contains an "A" or "a"
   {
       my $numE = $word =~ tr/eE//;
       print "$word $numE\n";  #number of e's in word with an "a"
   }
}

__END__
anyone 1
cancel 1
declArE 2
pal 0
[download]

I think shortening the code to a one-liner is a meaningless exercise.

Update: I don't see the reason to make a "one liner" other than to do it for the fun of doing it. This makes no difference in execution speed. White space and comments consume no MIPS and the difference in Perl compile speed is insignificant. This $words[$count] is very anti-Perl philosophy... subscripts are rare. Learn more Perl before trying to write a one-liner.

Another Update:
This will be about the same performance as the above code:

my @words = ("anyone", "cancel", "declArE", "perlmonks", "pal");
print "$_: ",tr/eE//, "\n" for grep{tr/aA//;}@words;
[download]

I asked a Python friend and here is one result:

words = ["anyone", "cancel", "declArE", "perlmonks", "pal"]
print(*((word,word.upper().count('E')) for word in words if 'A' in wor
+d.upper()))
[download]

I like my Perl code much better!