If your normal level of employment has been established as one and two days respectively, then the Department are correct in saying that if you work at this level you are not entitled to a Jobseekers Benefit claim. The reason for this is that one of the conditions is that you must have lost at least one day's employment to qualify.
You should ask for the calculation of how they got the one and two days. They usually use the last 13 weeks as a reference period, but you can argue that 26 or 52 weeks should be used if it is more representative of your work over the past year. It does seem odd that they have come to a different conclusion for you and your OH given that you have worked the same days (do you both have 260+ contributions over your working life? Those with less than 260 contributions only get JB for 9 months whereas those with more get 12 months).
If using 26 or 52 weeks wouldn't make a difference, then your appeal is probably doomed to failure. If it would make a difference, ask for a review of your claim. They will tell you that they can't review it because it has gone to appeal, but they are supposed to review your file anyway when you submit an appeal, so insist that this is done and make representations, in writing, that you want a different period used for the calculation. Of course, to do this you first need a written statement of how the thing was calculated in the first place. They should provide this to you, if they don't then you can request your file from the Department under the Freedom of Information office.
For any weeks that you are under the one or two days employment and therefore eligible for a payment, if they have closed your file, then you can simply submit a new Jobseekers Benefit application, although it should link to your previous claim if there hasn't been more than 26 weeks since that.