I believe that EE requires gzip_output to be set and not compress_output. There are several places in the EE system files that are using/referencing gzip_output but make no mention of compress_output, most tellingly in Template.php. Compress_output may be limited to just CodeIgniter itself.
expressionengine/libraries/Template.php
2869 // GZip mode: {gzip_mode}
2870: $str = str_replace(LD.'gzip_mode'.RD, ($this->EE->config->item('gzip_output') == 'y') ? $this->EE->lang->line('enabled') : $this->EE->lang->line('disabled'), $str);
2871
2872 // App version: {version}
expressionengine/models/admin_model.php
75 'output_cfg' => array(
76 'send_headers' => array('r', array('y' => 'yes', 'n' => 'no')),
77: 'gzip_output' => array('r', array('y' => 'yes', 'n' => 'no')),
78 'force_query_string' => array('r', array('y' => 'yes', 'n' => 'no')),
79 'redirect_method' => array('s', array('redirect' => 'location_method', 'refresh' => 'refresh_method')),
258 'template_debugging' => array('template_debugging_explanation'),
259 'max_caches' => array('max_caches_explanation'),
260: 'gzip_output' => array('gzip_output_explanation'),
261 'server_offset' => array('server_offset_explain'),
262 'default_member_group' => array('group_assignment_defaults_to_two'),