Yes, you can run PHP via SSH on a Jelastic nginx PHP node.
There's a tutorial at http://kb.layershift.com/jelastic-install-composer describing how to install composer on Apache and nginx PHP nodes on the Jelastic platform. I think it will help with your issue.
EDIT: I tested PHP CLI via SSH on a Jelastic nginx PHP node and confirmed it works fine for all PHP versions:
nginx@nginxphp /var/www $ which php
/usr/bin/php
nginx@nginxphp /var/www $ echo $PATH
/usr/local/bin:/bin:/usr/bin:/usr/local/sbin:/usr/sbin:/sbin
nginx@nginxphp /var/www $ php -v
PHP 5.3.29 (cli) (built: Aug 24 2014 19:41:55)
Copyright (c) 1997-2014 The PHP Group
Zend Engine v2.3.0, Copyright (c) 1998-2014 Zend Technologies
nginx@nginxphp /var/www $ php -v
PHP 5.4.32 (cli) (built: Sep 4 2014 10:33:47)
Copyright (c) 1997-2014 The PHP Group
Zend Engine v2.4.0, Copyright (c) 1998-2014 Zend Technologies
nginx@nginxphp /var/www $ php -v
PHP 5.5.16 (cli) (built: Sep 4 2014 10:19:16)
Copyright (c) 1997-2014 The PHP Group
Zend Engine v2.5.0, Copyright (c) 1998-2014 Zend Technologies
nginx@nginxphp /var/www $ php -v
PHP 5.6.0 (cli) (built: Sep 4 2014 11:18:01)
Copyright (c) 1997-2014 The PHP Group
Zend Engine v2.6.0, Copyright (c) 1998-2014 Zend Technologies
The above is whilst switching the PHP version via the Jelastic dashboard, and is taken using the latest Jelastic version (2.5). If you don't have the same behaviour, I think you should contact your hosting provider for clarification because it should work fine.
Composer install completes ok for me just like this:
nginx@nginxphp ~ $ curl -sS https://getcomposer.org/installer | php
#!/usr/bin/env php
All settings correct for using Composer
Downloading...
Composer successfully installed to: /var/lib/nginx/composer.phar
Use it: php composer.phar