No, they are not eligible. 

To be eligible to claim, receive and continue receiving loss of employment insurance benefits, the customer must have been employed and have been continuously working at their principal occupation for a minimum of 20 hours per week for 12 consecutive months as of the effective date of their voluntary loss of employment.

For example, if a customer has not worked for 2 weeks prior to staring a job at a new company, they will not be eligible to claim.