The thats mark these sentences as 'indirect' speech, in which the speaker who utters the sentence reports the substance of a prior utterance in a subordinate clause headed by that.
The subordinate clause is conventionally cast in the same tense as the main clause:
pres me that John is
past me that John was
pres that he will
pres not work.
past that he would
past not work. (Note that in this pair the auxiliary verbs will/would indicate future reference, but are themselves grammatically cast in the present-tense form. English has no future-tense forms.)
HOWEVER—If the substance reported (the content of the that clause) still holds at the time it is reported, you are free to cast the subordinate clause in the present tense. For instance:
I just saw
past Bill and he told
past me that John is
pres busy right now but will
pres be free this afternoon.
past Bill yesterday and he told
past me that he will
pres not work tomorrow because he has to meet a client.