Learn Farsi: اگر بک‌آپ نگرفته بودید، سرور هنگ می‌کرد. - If you had not taken a backup, the server would have frozen.

We use the negative prefix نـ (na-) on the past participle گرفته (gerefte), followed by the past tense of the verb to be, بودید (budid). Notice that the negative prefix attaches to the main verb, not to the auxiliary verb بودید.

Why does the result clause use هنگ می‌کرد instead of a perfect tense for 'would have frozen'?

In Farsi third conditionals (talking about unreal situations in the past), the result clause usually takes the past continuous tense. This is formed with the prefix می (mi-) plus the simple past of the verb. So while English uses the complex 'would have frozen', Farsi simply uses the past continuous form of هنگ کردن.

Are the words بک‌آپ, سرور, and هنگ really just English loanwords?

Exactly. Farsi uses English loanwords heavily for modern technology. Notice how Farsi turns them into actions by using light verbs: بک‌آپ گرفتن (to take a backup) uses گرفتن (to take), and هنگ کردن (to hang/freeze) uses کردن (to do).

Which syllable gets the stress in the negative verb نگرفته?

The negative prefix نـ (na-) always pulls the primary stress. You should emphasize the very beginning of the word.