can I say しごとがんばってね for saying have a nice day at work? do I need the particle と to make it "with work" しごととがんばってね
Japanese がんばる is a transitive verb that means "to work hard on/with ～". That is to say, you have to say しごとをがんばってね if you don't want to omit particles, but しごとがんばってね is fine in casual conversations, too. しごととがんばってね is ungrammatical.
In general, と meaning with cannot be easily omitted like が/は/を. Being able to omit と freely would obviously introduce a lot of confusion and ambiguity. For example, 彼【かれ】映画【えいが】見た【みた】 will always mean "He watched a movie" rather than "I watched a movie with him."