Does it make sense to train a CNN as an autoencoder?
Yes, it makes sense to use CNNs with autoencoders or other unsupervised methods. Indeed, different ways of combining CNNs with unsupervised training have been tried for EEG data, including using (convolutional and/or stacked) autoencoders.

Examples:

Deep Feature Learning for EEG Recordings uses convolutional autoencoders with custom constraints to improve generalization across subjects and trials.

EEG-based prediction of driver's cognitive performance by deep convolutional neural network uses convolutional deep belief networks on single electrodes and combines them with fully connected layers.

A novel deep learning approach for classification of EEG motor imagery signals uses fully connected stacked autoencoders on the output of a supervisedly trained (fairly shallow) CNN.  

But also purely supervised CNNs have had success on EEG data, see for example:

EEGNet: A Compact Convolutional Network for EEG-based Brain-Computer Interfaces

Deep learning with convolutional neural networks for brain mapping and decoding of movement-related information from the human EEG (disclosure: I am the first author of this work, more related work see p. 44)

Note that the EEGNet paper shows that also with a smaller number of trials, purely supervised training of their CNN can outperform their baselines (see Figure 3). Also in our experience on a dataset with only 288 training trials, purely supervised CNNs work fine, slightly outperforming a traditional filter bank common spatial patterns baseline.