Hi B_A_D. Sorry to hear about job loss and uncertainty about what is next.
Given how long you've been paying into EI, you most likely have coverage. Service Canada uses the submitted ROEs to determine eligibility and duration/terms for EI. Is it possible that your former employers already electronically submitted your ROEs but didn't give you a copy? If this is the case, you should be able to apply for EI online. You'll need your start and end dates of employment, SIN and bank account number for direct deposits.
With EI there is something called the Seven Day Rule
which states: "When an employee has had or is anticipated to have seven consecutive calendar days with no work and no insurable earnings from the employer, an interruption of earnings occurs." And with this interruption of earnings an ROE must be submitted to the Federal Government. There are exceptions to this rule, including non-standard work schedules.
Here's a link with the details:http://www.esdc.gc.ca/en/reports/ei/roe ... .page#s1_6
You have employee rights to know where you stand with your employers and they have legal obligations under the law, including submitting an ROE for interruption of earnings.
Hope this information helps you navigate the next steps.