Results

I was very impressed with the variety of methods, not to mention the ingenuity displayed! Without further ado, the results:

Non-Regex (7 entrants):

And the winner is.... Degausser (aka Jamie)!

1. Degausser (158 bytes)

Code:
alias palicount {
  var %x 99,%c 0
  while %x {
    var %s %s $+ $mid($1,%x,1),%y $len(%s)
    while %y > 2 {
      if (* $+ $mid($1,%x,%y) iswmcs %s) inc %c
      dec %y
    }
    dec %x
  }
  return %c
}


2. Sat (167 bytes)

Code:
alias palicount {
  var %i 2.9, %t 0
  while %i < $len($1) {
    var %j -.1
    while ($mid(#$1,$calc(%i -%j),1) === $mid($1,$calc(%i +%j),1)) inc %j
    inc %t $or(%j)
    inc %i .5
  }
  return %t
}


3. maroon (172 bytes)

Code:
alias palicount {
  var %c 0,%p 97
  while %p {
    var %o %p,%r
    while $mid($1,%o,1) != $5 {
      %r = $v1 $+ %r
      if ???* iswm %r && (%r $+ * iswmcs $mid($1,%p)) inc %c
      inc %o
    }
    dec %p
  }
  return %c
}


4. Tefad (189 bytes)

Code:
alias palicount {
  var %s $len($1),%i %s,%o 0
  while %i {
    var %d 2
    while %d {
      var %c %d - 1,%h %i + %c,%l %i - 1
      while %l && %h <= %s && $mid($1,%h,1) === $mid($1,%l,1) {
        inc %o %c
        inc %h
        dec %l
        %c = 1
      }
      dec %d
    }
    dec %i
  }
  return %o 
}


5. Kol (190 bytes)

Code:
alias palicount {
  var %x 99,%r 0
  while %x {
    var %y 99
    while %y { 
      var %t $mid($1@,%x,%y),%z 99,%c
      while %z {
        %c = %c $+ $mid(%t,%z,1)
        dec %z
      }
      if $poscs(%t %t,%c,2) > 4 {
        inc %r
      }
      dec %y
    }
    dec %x
  }
  return %r
}


6. Talon (209 bytes)

Code:
alias palicount {
  var %a 0,%d 0
  while %a < $len($1) {
    var %b 3,%c $v2 - %a
    inc %a
    while %b <= %c {
      var %y $mid($1,%a,%b),%z $len(%y),%x
      while %z { 
        %x = %x $+ $mid(%y,%z,1)
        dec %z
      }
      if (%y === %x) inc %d
      inc %b
    }
  }
  return %d
}


7. FroggieDaFrog (273 bytes)

Code:
alias -l c {
  var %o 
  while ($calc($2 -%o) > 0 && $mid($1,$v1,1) === $mid($1,$calc($2 +%o +$3),1)) {
    inc %o
  }
  return $calc(%o -1)
}
alias palicount {
  var %i 1,%r 0
  while (%i < $len($1)) {
    inc %r $c($1,%i)
    if ($mid($1,%i,1) === $mid($1,$calc(1+%i),1)) inc %r $c($1,%i,1)
    inc %i
  }
  return %r
}


Regex (2 entrants):

And the winner is.... maroon!

1. maroon (173 bytes)

Code:
alias palicount {
  var %c 0,%p 97
  while %p {
    var %o %p,%r
    while $mid($1,%o,1) != $5 {
      %r = $v1 $+ %r
      if $regex($mid($1,%p),^ $+ %r) && (???* iswm %r) inc %c
      inc %o
    }
    dec %p
  }
  return %c
}


2. Raccoon (214 bytes)

Code:
ALIAS palicount return $calc($regsubex($x($1),/(.)(?=(.*)$)/g,$y($x(\1\2)) +))
ALIAS x return $regsubex(x,$1,/(.)/g,$mid(\A,-\n,1))
ALIAS y return $len($regsubex(y,$1,/(.)(?=(.+\1)$)|./gF,$iif(\1\2 === $$x(\1\2),1)))


Raccoon should get accolades for having a far more regexy method smile

Thank you to everyone for competing!

"The only excuse for making a useless script is that one admires it intensely" - Oscar Wilde