Learn English – Is it correct to say: It is the brother’s car. or should I say: Is the brother’s car

All of these are to some degree acceptable:

1a.  It is important that we say "no" . . .
1b. ^?It is important we say "no" . . .
2a.  It is important that we should say "no" . . .
2b. ^?It is important we should say "no" . . .

The two 'b' versions, with no that, are entirely acceptable in sub-formal uses, and are usually acceptable in formal use; but the that makes the syntax clearer.

In 1a and 1b the verb takes what is traditionally called the mandative subjunctive - the unmarked infinitive form of the verb uninflected for person or number, employed in a subordinate clause expressing something desired or required. (We only know that it is the infinitive rather than the simple present because with the verb be, which is the only verb which distinguishes these two forms, the infinitive form is used.)

In 2a and 2b, the same 'subjunctive' quality is expressed using the past form of the modal verb shall, which itself of course bears a 'mandative' sense.

Both uses are acceptable, and have been for centuries; but that with the modal verb has been gaining ground over time.

^✲ marks an utterance as unacceptable ^? marks an utterance as possibly unacceptable

This is an ongoing awkwardness in the English language. It still sounds awkward to many of us to use the third person plural to mean a third person masculine-or-feminine singular. However, I think it will eventually be adopted, simply because saying "him or her", "he or she", "him- or herself" (I prefer to use the hyphen) is even more awkward.

Note that in the original sentence "him or her self" is incorrect (hard to notice because the line break occurs there). Herself is one word.