这个句子不对, 不仅仅是everyone前面加不加and的问题,还有语序及结构错误.

we must know 后跟的是宾语从句, 宾从中应用陈述语序,
而what should we done是疑问语序.所以错了.

另外, should worked 也是错的, 情态动词后应跟动词原形.

改正后的句子为:
we must know what we should do,and everyone should work hard.
句意:我们必须知道我们应当做什么,而且大家应该努力工作.

Everyone should do something he like. 这个句子中的 like应改为 likes,因为主语he为单三.

Everyone should do something he likes.