Is there a way to squash a number of commits non-interactively?

Make sure your working tree is clean, then

git reset --soft HEAD~3
git commit -m 'new commit message'

To add to the answer by wilhelmtell I find it convenient to soft reset to HEAD~2 and then amending the commit of HEAD~3:

git reset --soft HEAD~2
git commit --all --amend --no-edit    

This will merge all commits to the HEAD~3 commit and use its commit message. Be sure to start from a clean working tree.

I personally like wilhelmtell's solution:

git reset --soft HEAD~3
git commit -m 'new commit message'

However, I made an alias with some error checking so that you can do this:

g.squash 3 'my commit message'

I recommend setting up aliases that actually run scripts so that it is easier to (a) code up your scripts and (b) do more complex work with error checking. Below is a script that does the work of squashing. I put that in a scripts folder in my HOME path.

Script for squashing (squash.sh)

#!/bin/bash
#

#get number of commits to squash
squashCount=$1

#get the commit message
shift
commitMsg=$@

#regular expression to verify that squash number is an integer
regex='^[0-9]+$'

echo "---------------------------------"
echo "Will squash $squashCount commits"
echo "Commit message will be '$commitMsg'"

echo "...validating input"
if ! [[ $squashCount =~ $regex ]]
then
    echo "Squash count must be an integer."
elif [ -z "$commitMsg" ]
then
    echo "Invalid commit message.  Make sure string is not empty"
else
    echo "...input looks good"
    echo "...proceeding to squash"
    git reset --soft HEAD~$squashCount
    git commit -m "$commitMsg"
    echo "...done"
fi

echo
exit 0

Then to hook up that squash.sh script to an alias, I add the following to my .zprofile:

export PATH="$PATH:$HOME/scripts" # Add scripts folder to PATH
...
alias g.squash='function _gSquash(){ sh squash.sh $1 $2; };_gSquash'
...

Note: You can make your alias anything you want. I have my a lot of my git shortcuts as g.<myCommand>