It would validate if you put it in {} braces:
{$today.day -eq 15} { Write-Host Greater than 13 }
BTW, the switch value (e.g $today.day) is available as $_ inside the switch statement, so this is valid:
{$_ -eq 15} { Write-Host Greater than 13 }
Further more, you can even do regex inside:
{$_ -match '\d{2}'} { Write-Host 'Greater than or equel to 10'}