If someone is eligible, they are single and available to go on a date. A looter is someone who steals, particularly during a disaster of some type when police are busy with other emergencies, such as during a power failure, or after an earthquake or hurricane. The daughter is sarcastically asking why she should be worried about her appearance during a blackout because the only men she's likely to meet are power company employees or criminals.

Can I just add something to Susan612's explanation of 'looter' (someone who steals, particularly during a disaster of some type when police are busy with other emergencies) ? 'Eligible looters' is a play on words based on the expression 'eligible suitors'. In the past, mothers would always encourage their daughters to go out looking their best in case they should meet an 'eligible suitor' - in other words, a nice single man who might want to marry her. If you have read or seen anything written by Jane Austen, you will know what we mean by an 'eligible suitor'. This joke is part of the daughter's sarcastic response. The daughter doesn't think it is worth worrying about her appearance in this situation, and she is mocking her mother for being concerned about it.