BULK INSERT runs fine in window, but fails inside sql job
When you run that query interactively then SQL Server impersonates you when accessing the share. Normally this would fall into double-hop Kerberos Constrained Delegation and fail unless set up appropriately. It works because probably one of the hops is short-circuited (eg. you log on from a session on Server A or Server B).

When the query is run as a job the query impersonates the SQL Server agent service, and it all depends on how the SQL Server Agent is configured to run and connect to SQL Server. Different answers apply depending on whether the connection is made using a SQL Server login (eg. sa), a domain account, a virtual account, a local built-in account (eg. LocalService) and so on and so forth.

We here cannot guess what the problem is. To document it, please provide the details of the accounts involved. Are Server A/B in a domain? In the same domain? Is SQL Server running as a domain account, as a virtual account, as a local account, as local service, or what? Is SQL Server Agent running as who?

When you granted access to Everyone, did you grant access to the share or did you grant access to the folder?

Ideally you should set up an audit policy and then simply consult the event log, which will tell you exactly which account attempted the access. See Auditing File Access on File Servers 

Finally, you may be interested in reading up about SQL Server Agent proxies.