I'm declaring bankruptcy on this. If anyone is invested in 3.7 support, they're welcome to pick this up. https://api.travis-ci.org/v3/job/398380615/log.txt is the last build I have of this.

As far as I can tell what is happening is that Python is correctly building against the right version of OpenSSL that I am telling it to but then at runtime is loading the shared library for the system OpenSSL. I have no idea what to do about this and I no longer care.

@DRMacIver I've got a way to break into travis worker box:

install:
- sudo apt-get install -y software-properties-common
- sudo add-apt-repository -y ppa:tmate.io/archive
- sudo apt-get update -y
- sudo apt-get install -y tmate

script:
- ssh-keygen -b 4096 -t rsa -f ~/.ssh/id_rsa -q -P ""
- tmate -S /tmp/tmate.sock new-session -d               # Launch tmate in a detached state
- tmate -S /tmp/tmate.sock wait tmate-ready             # Blocks until the SSH connection is established
- tmate -S /tmp/tmate.sock display -p '#{tmate_ssh}'    # Prints the SSH connection string
- tmate -S /tmp/tmate.sock display -p '#{tmate_ssh_ro}' # Prints the read-only SSH connection string
- tmate -S /tmp/tmate.sock display -p '#{tmate_web}'    # Prints the web connection string
- tmate -S /tmp/tmate.sock display -p '#{tmate_web_ro}' # Prints the read-only web connection string
- |
    while :
    do
      echo -n .
      sleep 30s
    done &
- sleep infinity

This might help you to explore more without having to push new commits. (Well, you can actually also use their docker images for testing locally)

excerpt from build log:

*** WARNING: renaming "_ssl" since importing it failed: build/lib.linux-x86_64-3.7/_ssl.cpython-37m-x86_64-linux-gnu.so: undefined symbol: SSL_CTX_set_alpn_protos

The following modules found by detect_modules() in setup.py, have been
built by the Makefile instead, as configured by the Setup files:
_abc                  atexit                pwd                
time                                                           


Following modules built successfully but were removed because they could not be imported:
_ssl                                                           


Could not build the ssl module!
Python requires an OpenSSL 1.0.2 or 1.1 compatible libssl with X509_VERIFY_PARAM_set1_host().
LibreSSL 2.6.4 and earlier do not provide the necessary APIs, https://github.com/libressl-portable/portable/issues/381

I was able to build it under travisci/ci-garnet:packer-1512502276-986baf0 (docker image used by Travis CI) by downloading and compiling openssl and then downloading and compiling cpython, with env vars pointing to dir with openssl sources and *.sos: LDFLAGS="-L$HOME/openssl-1.1.0h" CFLAGS="-I$HOME/openssl-1.1.0h"