Due to the fact that Packer is written in Go programming language, to install Packer, you only need to download the binary from their website found at https://www.packer.io/downloads.html. You can usually do something like the following for a quick installation:
$ # Get the archive
$ wget -q --show-progress https://releases.hashicorp.com/packer/1.1.1/packer_<release>.zip
packer_<release>.zip 100%[==============================================>] 15.80M 316KB/s in 40s
$ # Extract our binary
$ unzip packer_<release>.zip
Archive: packer_<release>.zip
inflating: packer
$ # Place the binary somewhere in your path
$ sudo mv packer /usr/local/bin/
$ packer --version
1.1.1
CAUTION! Packer binaries only provide TLS authentication for their runner without any form of signature checking, so the assurance that the program was published by HashiCorp itself is orders of magnitude lower than a GPG-signed apt repository that Docker uses; so, exercise extra care when getting it this way or build it from source (https://github.com/hashicorp/packer).