正确：The old worker insisted that he wasn't old and went back to the working post again.
主语是the old worker,并列谓语insisted和went (back),insisted带一个从句：从句主系表结构,he wasn't old
went back写成back勉强可以,很不讲究,back作动词表后退主要指倒退的动作而不是状态,例如马向后退了两步：the horse backed steps.
正确：The old worker insisted that he wasn't old and should be sent back to the working post.
主语是the old worker,谓语是insisted,带一个并列结构的从句：从句主语he,第一部分系表结构,wasn't old,第二部分情态动词构成的被动语态,should be sent back to the working post again
这句等于：
The old worker insisted that he wasn't old and (insisted that he) should be sent back to the working post.
不要听别人瞎说,这里的should是不可以省略的.
错误：The old worker insisted that he wasn't old ,and be sent back to the working post again.
主句主语the old worker,从句主语he,不能直接接be sent back.
被动语态的主语如果是人,一般表达一种状态,某某被怎样怎样了,某某将被怎样怎样,或某某正在被怎样怎样.insist坚持,后用被动语态的事情：一,不能是过去的事情；二,不可能正在进行；三,施事者是别人也无法保证将来一定能够实现,因此只能是应该被怎样表将来可能的一种状态.