"Do you have to go to eat?"
Translation:क्या आपको खाने जाना है?
These two forms are not found in similar situations.
verb + ना is used when the verb is in the subjunctive mood. This is used to represent things like has to do / should do etc. e.g. मैं खाता हूँ। - I eat. मुझे खाना है। - I have/want to eat.
verb + ने represents that the verb is in the infinitive mood. It is part of a larger phrase. This usually happens when this is the main verb and has a helping verb or it is the object of a postposition (exactly like how it is in English). It is equivalent to 'to verb'. e.g. खाने - to eat. खाने के लिए - to eat. खाने से - from/(due to) eating. खाने जा - go to eat. खाने आ - come to eat.
And also like English when the verb phrase has a main verb and a helping verb, the main verb is in the infinitive/participle/gerund form and all the inflection is transferred to the helping verb. The only difference being that in English the helping verb is written first, while in Hindi the main verb is written first.
In this sentence, खा is the main verb and जा is the helping verb (Just like in English 'to go to eat': 'go' is the helping verb and 'eat' is the main verb). So, खा will be in the infinitive खाने and जा will be conjugated according to the needs of the sentence. Since the sentence is in the present tense subjunctive mood, it will be क्या आपको खाने जाना है? If the sentence had been in present indicative it would be क्या आप खाने जाते हैं? In third person - क्या वह खाने जाता है?